Capacitor

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Figure 3.3 Capacitor.

The capacitor stores charge and the relationship between the charge stored and the resultant voltage is q = Cv. The constant of proportionality, the capacitance, has units of farads (F), and is named for the English experimental physicist Michae l Faraday. As current is the rate of change of charge, the v-i relation can be expressed in differential or integral form.

$i(t)=C\frac{d}{dt}v(t)\ or\ v(t)=\frac{1}{C}\int_{-\infty }^{t}i(\alpha)d\alpha$

If the voltage across a capacitor is constant, then the current flowing into it equals zero. In this situation, the capacitor is equivalent to an open circuit. The power consumed/produced by a voltage applied to a capacitor depends on the product of the voltage and its derivative.

$p(t)=Cv(t)\frac{d}{dt}v(t)$

This result means that a capacitor's total energy expenditure up to time t is concisely given by

$E(t)=\frac{1}{2}Cv^2(t)$

This expression presumes the fundamental assumption of circuit theory: all voltages and currents in any circuit were zero in the far distant past (t = −∞).

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Capacitor